Question

4. Factor the trinomial completely.
Type your answers to each problem in the table below. The order of the factors doesn't matter, but use parentheses around each one! 9x^2 - 9x - 4

Answer 1

To factor the trinomial \(9x^2 - 9x - 4\) completely, we can use the method of factoring by grouping or the quadratic formula to find the roots.

We'll start with factoring:

1. **Identify** \(a\), \(b\), and \(c\) in the trinomial \(ax^2 + bx + c\):
- \(a = 9\)
- \(b = -9\)
- \(c = -4\)

2. **Calculate the discriminant** \(D = b^2 - 4ac\):
\[
D = (-9)^2 - 4(9)(-4) = 81 + 144 = 225
\]
Since the discriminant is a perfect square, we can factor it.

3. **Find the roots using the quadratic formula**:
\[
x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm 15}{18}
\]
- \(x_1 = \frac{24}{18} = \frac{4}{3}\)
- \(x_2 = \frac{-6}{18} = -\frac{1}{3}\)

4. **Write the factored form using the roots**:
The factors corresponding to the roots are \(x - \frac{4}{3}\) and \(x + \frac{1}{3}\).

5. **Adjusting the factors to eliminate the fractions**:
Multiply each factor by 3:
\[
3 \left( x - \frac{4}{3} \right) = 3x - 4,\quad 3 \left( x + \frac{1}{3} \right) = 3x + 1
\]

Thus, the factored form becomes:
\[
(3x - 4)(3x + 1)
\]

Putting this in the requested format, the answer is:
\((3x - 4)(3x + 1)\)